Related papers: Some results on chromatic number as a function of …

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$…

Let $G$ be a claw-free graph on $n$ vertices with clique number $\omega$, and consider the chromatic number $\chi(G^2)$ of the square $G^2$ of $G$. Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the line graphs $L$ of simple…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A classic result of Vizing indicates that either $\chi'(G )=\Delta(G)$ or $\chi'(G )=\Delta(G)+1$. The graph $G$ is called $\Delta$-critical if $G$ is…

An edge coloring of a graph $G$ is called conflict-free if, for every edge, its closed neighborhood contains a color that appears exactly once. The least number of colors required for such a coloring is the conflict-free chromatic index of…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

The maximum number of edges in a graph with matching number m and maximum degree d has been determined in [1] and [2], where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is…

Let $A\subset\mathbb{R}_{>0}$ be a finite set of distances, and let $G_{A}(\mathbb{R}^{n})$ be the graph with vertex set $\mathbb{R}^{n}$ and edge set $\{(x,y)\in\mathbb{R}^{n}:\ \|x-y\|_{2}\in A\}$, and let…

The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a…

The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le…

We prove that for an arbitrary $\varepsilon > 0$ holds \[ \chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where $\chi(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of…

For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies…

Problem of finding an optimal upper bound for the chromatic no. of 3K1-free graphs is still open and pretty hard. It was proved by Choudum et al that an upper bound on the chromatic no. of {3K1, K1+C4}-free graphs, is 2{\omega}. We improve…

The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$ in terms…

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborova and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation…

Let $G$ be an $n$-vertex triangle-free graph. The celebrated Mantel's theorem showed that $e(G)\leq \lfloor\frac{n^2}{4}\rfloor$. In 1962, Erd\H{o}s (together with Gallai), and independently Andr\'{a}sfai, proved that if $G$ is…

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\mathcal{G}$ is said to be {\it$\chi$-bounded} if there exists…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

For a graph $G$, let $\chi(G)$ ($\omega(G)$) denote its chromatic (clique) number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the complement…

Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $\delta^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an…